# Parsimony-Enhanced Sparse Bayesian Learning for Robust Discovery of Partial Differential Equations

arXiv: https://arxiv.org/abs/2107.07040 · HTML: https://mlsanity.com/paper/2107.07040/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 46 references that are in the map.
- First posted on arXiv 8 Jul 2021.

## Builds on (most cited first, 16 of 16)

- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [DeepMoD: Deep learning for Model Discovery in noisy data](https://mlsanity.com/paper/1904.09406/) (2019)
- [DLGA-PDE: Discovery of PDEs with incomplete candidate library via combination of deep learning and genetic algorithm](https://mlsanity.com/paper/2001.07305/) (2020)
- [Data-driven discovery of PDEs in complex datasets](https://mlsanity.com/paper/1808.10788/) (2018)
- [DL-PDE: Deep-learning based data-driven discovery of partial differential equations from discrete and noisy data](https://mlsanity.com/paper/1908.04463/) (2019)
- [On spike-and-slab priors for Bayesian equation discovery of nonlinear dynamical systems via sparse linear regression](https://mlsanity.com/paper/2012.01937/) (2020)
- [Gaussian Process Assisted Active Learning of Physical Laws](https://mlsanity.com/paper/1910.03120/) (2019)
- [Gaussian processes meet NeuralODEs: A Bayesian framework for learning the dynamics of partially observed systems from scarce and noisy data](https://mlsanity.com/paper/2103.03385/) (2021)
- [Learning Partial Differential Equations from Data Using Neural Networks](https://mlsanity.com/paper/1910.10262/) (2019)
- [Data-driven Identification of 2D Partial Differential Equations using extracted physical features](https://mlsanity.com/paper/2010.10626/) (2020)
- [Data-driven PDE discovery with evolutionary approach](https://mlsanity.com/paper/1903.08011/) (2019)
- [Machine Learning of Partial Differential Equations from Noise Data](https://mlsanity.com/paper/2010.06507/) (2020)
- [Machine Discovery of Partial Differential Equations from Spatiotemporal Data](https://mlsanity.com/paper/1909.06730/) (2019)
- [SubTSBR to tackle high noise and outliers for data-driven discovery of differential equations](https://mlsanity.com/paper/1907.07788/) (2019)

## Built on by (most cited first, 3 of 3)

- [Physics-constrained robust learning of open-form partial differential equations from limited and noisy data](https://mlsanity.com/paper/2309.07672/) (2023)
- [Physics-informed AI and ML-based sparse system identification algorithm for discovery of PDE's representing nonlinear dynamic systems](https://mlsanity.com/paper/2410.10023/) (2024)
- [Towards true discovery of the differential equations](https://mlsanity.com/paper/2308.04901/) (2023)

## Abstract (arXiv)

Robust physics discovery is of great interest for many scientific and engineering fields. Inspired by the principle that a representative model is the one simplest possible, a new model selection criteria considering both model's Parsimony and Sparsity is proposed. A Parsimony Enhanced Sparse Bayesian Learning (PeSBL) method is developed for discovering the governing Partial Differential Equations (PDEs) of nonlinear dynamical systems. Compared with the conventional Sparse Bayesian Learning (SBL) method, the PeSBL method promotes parsimony of the learned model in addition to its sparsity. In this method, the parsimony of model terms is evaluated using their locations in the prescribed candidate library, for the first time, considering the increased complexity with the power of polynomials and the order of spatial derivatives. Subsequently, the model parameters are updated through Bayesian inference with the raw data. This procedure aims to reduce the error associated with the possible loss of information in data preprocessing and numerical differentiation prior to sparse regression. Results of numerical case studies indicate that the governing PDEs of many canonical dynamical systems can be correctly identified using the proposed PeSBL method from highly noisy data (up to 50% in the current study). Next, the proposed methodology is extended for stochastic PDE learning where all parameters and modeling error are considered as random variables. Hierarchical Bayesian Inference (HBI) is integrated with the proposed framework for stochastic PDE learning from a population of observations. Finally, the proposed PeSBL is demonstrated for system response prediction with uncertainties and anomaly diagnosis. Codes of all demonstrated examples in this study are available on the website: https://github.com/ymlasu.